//==================================================================================================
/*
  EVE - Expressive Vector Engine
  Copyright : EVE Project Contributors
  SPDX-License-Identifier: BSL-1.0
*/
//==================================================================================================
#pragma once

#include <eve/arch.hpp>
#include <eve/traits/overload.hpp>
#include <eve/detail/hz_device.hpp>
#include <eve/module/core.hpp>
#include <eve/module/math.hpp>
#include <eve/module/special/regular/zeta.hpp>
//#include <eve/module/special/regular/trigamma.hpp>

namespace eve
{
  template<typename Options>
  struct polygamma_t : callable<polygamma_t, Options>
  {
    template<eve::scalar_value N, eve::floating_value T>
    constexpr EVE_FORCEINLINE T operator()(N n, T v) const  { return EVE_DISPATCH_CALL(n, v); }

    EVE_CALLABLE_OBJECT(polygamma_t, polygamma_);
  };

//================================================================================================
//! @addtogroup special
//! @{
//!   @var polygamma
//!   @brief `elementwise_callable` object computing the Polygamma function i.e.
//!   the nth derivative of the digamma function
//!
//!   @groupheader{Header file}
//!
//!   @code
//!   #include <eve/module/special.hpp>
//!   @endcode
//!
//!   @groupheader{Callable Signatures}
//!
//!   @code
//!   namespace eve
//!   {
//!      // Regular overload
//!      constexpr auto polygamma(scalar_value auto n, floating_value auto x)                          noexcept; // 1
//!
//!      // Lanes masking
//!      constexpr auto polygamma[conditional_expr auto c](scalar_value auto n, floating_value auto x) noexcept; // 2
//!      constexpr auto polygamma[logical_value auto m](scalar_value auto n, floating_value auto x)    noexcept; // 2
//!   }
//!   @endcode
//!
//!   **Parameters**
//!
//!     * `n`: [scalar_value](@ref eve::value). (see note below)
//!     * `x`: [flaoting value](@ref eve::floating_value).
//!     * `c`: [Conditional expression](@ref eve::conditional_expr) masking the operation.
//!     * `m`: [Logical value](@ref eve::logical_value) masking the operation.
//!
//!   **Return value**
//!
//!     1. The value of the Polygamma function: \f$\psi(n, x) = \frac{\mbox{d}^{n+1}}{\mbox{d}x^{n+1}}\log\Gamma(x)\f$ is returned.
//!     2. [The operation is performed conditionnaly](@ref conditional).
//!
//!  @note Only integral (or flint) positive values of n correspond to the proper polygamma definition. You can use
//!        non integral values but the function result may be different to
//!        the standard definitions of polygamma extensions.
//!
//!  @groupheader{External references}
//!   *  [DLMF: Gamma and Psi Functions](https://dlmf.nist.gov/5.2#i)
//!   *  [Wolfram MathWorld: Polygamma Function](https://mathworld.wolfram.com/PolygammaFunction.html)
//!
//!   @groupheader{Example}
//!   @godbolt{doc/special/polygamma.cpp}
//================================================================================================
  inline constexpr auto polygamma = functor<polygamma_t>;
//================================================================================================
//! @}
//================================================================================================

  namespace detail
  {
    template<typename N, typename T, callable_options O>
    constexpr T  polygamma_(EVE_REQUIRES(cpu_), O const&, N m, T z) noexcept
    {
      if (is_eqz(m)) return eve::digamma(z);
//      if (m == 1)    return eve::trigamma(z);
//      auto pi  = eve::pi(eve::as(z));
      using t_t = underlying_type_t<T>;

      constexpr t_t coefs[] =
        {t_t(0x1p+0),
         t_t(0x1p+0), t_t(0x1p+0),
         t_t(0x1p+0), t_t(0x1p+0),
         t_t(0x1.5555555555555p-2), t_t(0x1.5555555555555p+0), t_t(0x1p+0),
         t_t(0x1.5555555555555p-1), t_t(0x1.aaaaaaaaaaaaap+0), t_t(0x1p+0),
         t_t(0x1.1111111111111p-3), t_t(0x1.2222222222222p+0), t_t(0x1p+1), t_t(0x1p+0),
         t_t(0x1.82d82d82d82d8p-2), t_t(0x1.b60b60b60b60bp+0), t_t(0x1.2aaaaaaaaaaabp+1), t_t(0x1p+0),
         t_t(0x1.ba1ba1ba1ba1bp-5), t_t(0x1.9319319319319p-1), t_t(0x1.3333333333333p+1), t_t(0x1.5555555555556p+1), t_t(0x1p+0),
         t_t(0x1.9319319319319p-3), t_t(0x1.6596596596596p+0), t_t(0x1.999999999999ap+1), t_t(0x1.8p+1), t_t(0x1p+0),
         t_t(0x1.664f4882c10fap-6), t_t(0x1.f32d6bba49dd7p-2), t_t(0x1.1f2747c9d1f27p+1), t_t(0x1.071c71c71c71cp+2), t_t(0x1.aaaaaaaaaaaabp+1), t_t(0x1p+0),
         t_t(0x1.8f578961d4b12p-4), t_t(0x1.fd5cfda2574d3p-1), t_t(0x1.ae983ed942e98p+1), t_t(0x1.4888888888888p+2), t_t(0x1.d555555555556p+1), t_t(0x1p+0),
         t_t(0x1.226e355e6c23cp-7), t_t(0x1.1ee91349518b4p-2), t_t(0x1.cce88a446001cp+0), t_t(0x1.32edd988432edp+2), t_t(0x1.9111111111111p+2), t_t(0x1p+2), t_t(0x1p+0),
         t_t(0x1.7e8c19b717645p-5), t_t(0x1.4b2e731e5c221p-1), t_t(0x1.7fbf45e8fdd9dp+1), t_t(0x1.a4d7a2cf824d8p+2), t_t(0x1.e0b60b60b60b5p+2), t_t(0x1.1555555555555p+2), t_t(0x1p+0),
         t_t(0x1.d6d3d0e157ddfp-9), t_t(0x1.3910084bda6b4p-3), t_t(0x1.4d67430f57bf2p+0), t_t(0x1.2c67aa08d9712p+2), t_t(0x1.17b425ed097b4p+3), t_t(0x1.1bbbbbbbbbbbbp+3), t_t(0x1.2aaaaaaaaaaaap+2), t_t(0x1p+0),
         t_t(0x1.65c92e0d8be85p-6), t_t(0x1.9364df84aabd8p-2), t_t(0x1.311e76c0a257bp+1), t_t(0x1.c068068068068p+2), t_t(0x1.6a7f529fd4a7fp+3), t_t(0x1.4aaaaaaaaaaaap+3), t_t(0x1.4p+2), t_t(0x1p+0),
         t_t(0x1.7da36452b75e3p-10), t_t(0x1.48ada6c839db7p-4), t_t(0x1.bf2a2c27b654bp-1), t_t(0x1.041c05c26dbc5p+2), t_t(0x1.4220664aa8eedp+3), t_t(0x1.cbfcbfcbfcbfcp+3), t_t(0x1.7d27d27d27d27p+3), t_t(0x1.5555555555555p+2), t_t(0x1p+0),
         t_t(0x1.48ada6c839db7p-7), t_t(0x1.d3b5069439f26p-3), t_t(0x1.be0f4e289b651p+0), t_t(0x1.a3aae87392157p+2), t_t(0x1.c08e2b04d26f4p+3), t_t(0x1.1eadeadeadeaep+4), t_t(0x1.b333333333332p+3), t_t(0x1.6aaaaaaaaaaaap+2), t_t(0x1p+0),
         t_t(0x1.3558248036743p-11), t_t(0x1.4efacea047df4p-5), t_t(0x1.1b05b207d66f9p-1), t_t(0x1.9b34a54707a08p+1), t_t(0x1.43df64bffe6c8p+3), t_t(0x1.303bd089d56a2p+4), t_t(0x1.5fe5fe5fe5fe6p+4), t_t(0x1.ecccccccccccbp+3), t_t(0x1.7ffffffffffffp+2), t_t(0x1p+0),
         t_t(0x1.29c27ec75c54bp-8), t_t(0x1.04e15ce0c09b3p-3), t_t(0x1.319581f755cccp+0), t_t(0x1.646bca6f636dp+2), t_t(0x1.e1fb147cb389bp+3), t_t(0x1.939e1d700fe48p+4), t_t(0x1.aa3e4e8f93a3dp+4), t_t(0x1.14fa4fa4fa4fap+4), t_t(0x1.9555555555554p+2), t_t(0x1p+0),
         t_t(0x1.f57d7734d1663p-13), t_t(0x1.4d7370403481cp-6), t_t(0x1.564360fb26273p-2), t_t(0x1.2ed59f86d48cfp+1), t_t(0x1.25f6940c38339p+3), t_t(0x1.5bd3a35e699e3p+4), t_t(0x1.06a82365c4953p+5), t_t(0x1.fe4e8f93a3e4dp+4), t_t(0x1.3555555555554p+4), t_t(0x1.aaaaaaaaaaaa9p+2), t_t(0x1p+0),
         t_t(0x1.0ac2c0335d34ap-9), t_t(0x1.1a2596ca863c3p-4), t_t(0x1.8da0af87b6136p-1), t_t(0x1.18988191333e6p+2), t_t(0x1.d169783019b2dp+3), t_t(0x1.e919fc2953e88p+4), t_t(0x1.5033944a55b66p+5), t_t(0x1.2e573ac901e56p+5), t_t(0x1.5777777777776p+4), t_t(0x1.bfffffffffffep+2), t_t(0x1p+0),
         t_t(0x1.967e18afcafaep-14), t_t(0x1.45a1163d66b62p-7), t_t(0x1.8ed8abd8db90bp-3), t_t(0x1.a576da089d0c5p+0), t_t(0x1.ec74db59a1035p+2), t_t(0x1.63ed5aca9f2c5p+4), t_t(0x1.50390575cb094p+5), t_t(0x1.a8151812ed458p+5), t_t(0x1.62fada58502fap+5), t_t(0x1.7b60b60b60b5ep+4), t_t(0x1.d555555555554p+2), t_t(0x1p+0),
         t_t(0x1.d9a47d709566p-11), t_t(0x1.2978834da7d61p-5), t_t(0x1.f009b7a8bf493p-2), t_t(0x1.9f9f602b1627p+1), t_t(0x1.9d1b91b1399d4p+3), t_t(0x1.0849691135d4fp+5), t_t(0x1.c543cc7ad7314p+5), t_t(0x1.0804d70edd049p+6), t_t(0x1.9d5e008ab355dp+5), t_t(0x1.a11111111110fp+4), t_t(0x1.eaaaaaaaaaaa9p+2), t_t(0x1p+0),
         t_t(0x1.497d8eea25259p-15), t_t(0x1.38fa78ff8e8dep-8), t_t(0x1.c2bcd4c9e35e6p-4), t_t(0x1.17f23c2bbed63p+0), t_t(0x1.828c6ff56dad8p+2), t_t(0x1.4d9f1743d5064p+4), t_t(0x1.7e9757e97d1d8p+5), t_t(0x1.2c48676f31219p+6), t_t(0x1.44f3cf3cf3cf3p+6), t_t(0x1.ddcc877321dcbp+5), t_t(0x1.c888888888887p+4), t_t(0x1.fffffffffffffp+2), t_t(0x1p+0),
         t_t(0x1.a14df6aa1367dp-12), t_t(0x1.33031b0bea8c9p-6), t_t(0x1.2aba1a5ed2fa1p-2), t_t(0x1.24b13cd3c0f91p+1), t_t(0x1.567150b6c3ccdp+3), t_t(0x1.04ccd0cd8af01p+5), t_t(0x1.0ed012509145ap+6), t_t(0x1.87ccc67ed4885p+6), t_t(0x1.8bcf3cf3cf3cdp+6), t_t(0x1.1249249249248p+6), t_t(0x1.f1c71c71c71c5p+4), t_t(0x1.0aaaaaaaaaaaap+3), t_t(0x1p+0),
         t_t(0x1.0b132d39a605p-16), t_t(0x1.28d173849c24fp-9), t_t(0x1.f062456a5a84p-5), t_t(0x1.65b0280b206d7p-1), t_t(0x1.1f891eca323b8p+2), t_t(0x1.2324cb6522a11p+4), t_t(0x1.8c65b699bfbc4p+5), t_t(0x1.77e70ffb1cfb2p+6), t_t(0x1.f83afcb472d76p+6), t_t(0x1.dd9b8cec01f33p+6), t_t(0x1.38fd8fd8fd8fcp+6), t_t(0x1.0e66666666665p+5), t_t(0x1.1555555555554p+3), t_t(0x1p+0),
         t_t(0x1.6d508e2d0ef26p-13), t_t(0x1.372ccf6675c81p-7), t_t(0x1.5d43ec8e5c7dcp-3), t_t(0x1.8b296597addfbp+0), t_t(0x1.0c31661e225dap+3), t_t(0x1.dde231dc34759p+4), t_t(0x1.25e2a8e5c5b2bp+6), t_t(0x1.005a3484f9e86p+7), t_t(0x1.4079399ce8ef1p+7), t_t(0x1.1db4e81b4e81ap+7), t_t(0x1.6329729729727p+6), t_t(0x1.24ccccccccccbp+5), t_t(0x1.1ffffffffffffp+3), t_t(0x1p+0),
         t_t(0x1.b0f72d3ee24ebp-18), t_t(0x1.164a93168b226p-10), t_t(0x1.0b5bdae9edb5bp-5), t_t(0x1.ba22ff422ea7cp-2), t_t(0x1.98d0b682f3c47p+1), t_t(0x1.dec8c5ece0ddcp+3), t_t(0x1.7c58ef2b4dcd8p+5), t_t(0x1.aa5619e5f01f5p+6), t_t(0x1.583296c91d7ffp+7), t_t(0x1.92d51cf1452abp+7), t_t(0x1.532bde1e97f4cp+7), t_t(0x1.90f2b9d6480efp+6), t_t(0x1.3c16c16c16c15p+5), t_t(0x1.2aaaaaaaaaaa9p+3), t_t(0x1p+0),
         t_t(0x1.3e0c15d09f02bp-14), t_t(0x1.3685bcd052272p-8), t_t(0x1.8e1245dbb8f83p-4), t_t(0x1.014b6cd2f25ddp+0), t_t(0x1.906415299edf1p+2), t_t(0x1.9b825e0f6ad7bp+4), t_t(0x1.26abd27c3f972p+6), t_t(0x1.2f44b810fa9a1p+7), t_t(0x1.c7a5e8c4e1e49p+7), t_t(0x1.f53a9efa19012p+7), t_t(0x1.8fc7d560e447p+7), t_t(0x1.c27f529fd4a7bp+6), t_t(0x1.5444444444442p+5), t_t(0x1.3555555555555p+3), t_t(0x1p+0),
         t_t(0x1.5ef2da474e5b4p-19), t_t(0x1.025ade8f82779p-11), t_t(0x1.1a900f8b43e59p-6), t_t(0x1.0994a5763f392p-2), t_t(0x1.179241fe53b91p+1), t_t(0x1.764f1e807fc64p+3), t_t(0x1.563b9995a1233p+5), t_t(0x1.bdd1e7cdacfd9p+6), t_t(0x1.a7f78f61b697ep+7), t_t(0x1.29bf6b100c9bap+8), t_t(0x1.34f1a03ba176cp+8), t_t(0x1.d4244cef79a21p+7), t_t(0x1.f7f529fd4a7f1p+6), t_t(0x1.6d55555555554p+5), t_t(0x1.4p+3), t_t(0x1p+0),
         t_t(0x1.1394209913b2cp-15), t_t(0x1.31b4c77d5765dp-9), t_t(0x1.bbc43cf9adf34p-5), t_t(0x1.44c4791292d3bp-1), t_t(0x1.1ed0c7cce8e12p+2), t_t(0x1.502bccc051f06p+4), t_t(0x1.147ef14ab5391p+6), t_t(0x1.4a23f17db46ap+7), t_t(0x1.23b4ddf07c85ep+8), t_t(0x1.809c77dc0ccebp+8), t_t(0x1.799d53d4e50aap+8), t_t(0x1.10718b01eeb2cp+8), t_t(0x1.18bd167c126bcp+7), t_t(0x1.8749f49f49f49p+5), t_t(0x1.4aaaaaaaaaaabp+3), t_t(0x1p+0),
         t_t(0x1.1c77df95c1c0cp-20), t_t(0x1.db92ead3d3df2p-13), t_t(0x1.25b2767ce3e38p-7), t_t(0x1.373b6fd085704p-3), t_t(0x1.71be5088d4b1fp+0), t_t(0x1.183521ec6c78ap+3), t_t(0x1.238b0358857fbp+5), t_t(0x1.b370cac244e9ep+6), t_t(0x1.dfae78c8b6949p+7), t_t(0x1.8bb2919f64e93p+8), t_t(0x1.eb88697a2f64dp+8), t_t(0x1.c9f05ddbf9acdp+8), t_t(0x1.3b562a01ea133p+8), t_t(0x1.379a244cef799p+7), t_t(0x1.a222222222222p+5), t_t(0x1.5555555555556p+3), t_t(0x1p+0),
         t_t(0x1.db92ead3d3df2p-17), t_t(0x1.29699c528b8b4p-10), t_t(0x1.e5344f209666ap-6), t_t(0x1.8eebe30461347p-2), t_t(0x1.8c7a3478a22dp+1), t_t(0x1.06708fcf5512ap+4), t_t(0x1.ea56d2ab2e5c8p+5), t_t(0x1.4f17e8beda5d7p+7), t_t(0x1.5680101bd6685p+8), t_t(0x1.08e4d9eb03311p+9), t_t(0x1.37054139cca2ep+9), t_t(0x1.13aaf004559aap+9), t_t(0x1.6b173e40b0d7cp+8), t_t(0x1.58a4b4f5fa0a4p+7), t_t(0x1.bdddddddddddep+5), t_t(0x1.6000000000001p+3), t_t(0x1p+0),
         t_t(0x1.cd299de4ae6bcp-22), t_t(0x1.b266c72d3b4a7p-14), t_t(0x1.2cd2966af598ap-8), t_t(0x1.64db951286b82p-4), t_t(0x1.dad4a2c6010cp-1), t_t(0x1.93fc147a39f83p+2), t_t(0x1.d9cec66721dabp+4), t_t(0x1.91365a2c4fa53p+6), t_t(0x1.f931b8736eb5p+7), t_t(0x1.e1780afdee3fap+8), t_t(0x1.5e6fd5854dc1fp+9), t_t(0x1.860de04e97193p+9), t_t(0x1.49aa9bf4bcb5bp+9), t_t(0x1.a00f2b9d6480ep+8), t_t(0x1.7befbefbefbefp+7), t_t(0x1.da7d27d27d27dp+5), t_t(0x1.6aaaaaaaaaaacp+3), t_t(0x1p+0),
         t_t(0x1.98d933ee55ebbp-18), t_t(0x1.1e523ff9b4c3p-11), t_t(0x1.04bf4f88b19d1p-6), t_t(0x1.de63083bfdd04p-3), t_t(0x1.0991bb447c83ap+1), t_t(0x1.89dcad2d98e48p+3), t_t(0x1.9e05f4cd47908p+5), t_t(0x1.40574b3f71abep+7), t_t(0x1.75c3b8bfb64c5p+8), t_t(0x1.4d9644465637ap+9), t_t(0x1.ca86ffae71644p+9), t_t(0x1.e517ff39f608fp+9), t_t(0x1.87c9864ad1482p+9), t_t(0x1.da9b7f0d4629ap+8), t_t(0x1.a18e38e38e38dp+7), t_t(0x1.f8p+5), t_t(0x1.7555555555557p+3), t_t(0x1p+0),
         t_t(0x1.75cde6563fed7p-23), t_t(0x1.8a211e52bbddbp-15), t_t(0x1.3021e394c5b7bp-9), t_t(0x1.915593952f825p-5), t_t(0x1.29138858123f8p-1), t_t(0x1.19b70c3f1107p+2), t_t(0x1.7173e70266c9bp+4), t_t(0x1.5f77d0b4467eap+6), t_t(0x1.f45fe25915018p+7), t_t(0x1.0fdc88c9c5222p+9), t_t(0x1.c834dbbabbfb5p+9), t_t(0x1.28f23681e43aap+10), t_t(0x1.2b4fdb97d270dp+10), t_t(0x1.cee92e2f96f32p+9), t_t(0x1.0d8e774ddb441p+9), t_t(0x1.c993193193192p+7), t_t(0x1.0b33333333333p+6), t_t(0x1.8000000000002p+3), t_t(0x1p+0),
         t_t(0x1.5e5653d7c36fcp-19), t_t(0x1.1113afd6982a3p-12), t_t(0x1.14011a7cec46bp-7), t_t(0x1.18ca42e647889p-3), t_t(0x1.5a067216f880ap+0), t_t(0x1.1d6dcd7c2355bp+3), t_t(0x1.4ef081b747f5ep+5), t_t(0x1.22bacd2e337cdp+7), t_t(0x1.7f0e494f5806p+8), t_t(0x1.8560f75ad9392p+9), t_t(0x1.3430ec9178945p+10), t_t(0x1.7d02257bb4363p+10), t_t(0x1.6eb41bf6665bep+10), t_t(0x1.0ffd62a13d8ecp+10), t_t(0x1.30fbacb51af6ap+9), t_t(0x1.f411566abc01p+7), t_t(0x1.1ad82d82d82d8p+6), t_t(0x1.8aaaaaaaaaaacp+3), t_t(0x1p+0),
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         t_t(0x1.04b24b297b798p-55), t_t(0x1.4019d5f57800ep-46), t_t(0x1.e96c66e24a501p-39), t_t(0x1.702c33e8d5d9ep-32), t_t(0x1.4c93098f41546p-26), t_t(0x1.9363752798031p-21), t_t(0x1.60c59547cfc1dp-16), t_t(0x1.d352accf14a5fp-12), t_t(0x1.e603807ee4cf1p-8), t_t(0x1.97c01c2321cc5p-4), t_t(0x1.19e1212dd064ep+0), t_t(0x1.469c6ea168e3ap+3), t_t(0x1.41897322da9a2p+6), t_t(0x1.10009d43e7744p+9), t_t(0x1.8f2ee1d47e131p+11), t_t(0x1.0017bab59682ap+14), t_t(0x1.2133a4ac712c1p+16), t_t(0x1.21120eac5eb01p+18), t_t(0x1.00f8971b56956p+20), t_t(0x1.97f93be9285d7p+21), t_t(0x1.222eb1f48b967p+23), t_t(0x1.72f02975208acp+24), t_t(0x1.ab15ef76a23a3p+25), t_t(0x1.bbbeef2c4a867p+26), t_t(0x1.a0acb5821a08p+27), t_t(0x1.61f847a281df1p+28), t_t(0x1.103ae3bf9acf2p+29), t_t(0x1.7b3251ac2798ep+29), t_t(0x1.de479e1fb94aep+29), t_t(0x1.10fd8f483897fp+30), t_t(0x1.19cab4a157b4ap+30), t_t(0x1.06ab0f3624ab5p+30), t_t(0x1.b9617c958f00ep+29), t_t(0x1.4d75cf7da3236p+29), t_t(0x1.c3b08ddc215f4p+28), t_t(0x1.1135eef480813p+28), t_t(0x1.25cb02190353bp+27), t_t(0x1.1734da82fb01p+26), t_t(0x1.d1a59f81bf904p+24), t_t(0x1.51a6093f3b218p+23), t_t(0x1.a4f6a9d71a597p+21), t_t(0x1.bc8ffd56f0bedp+19), t_t(0x1.85e81f9bea3aap+17), t_t(0x1.1459b1423094dp+15), t_t(0x1.3024fea53fa96p+12), t_t(0x1.e7aaaaaaaaaaap+8), t_t(0x1.faaaaaaaaaaa9p+4), t_t(0x1p+0),
         t_t(0x1.5f4111bbef1fbp-62), t_t(0x1.43a4548e24a1dp-51), t_t(0x1.9d68dc894c309p-43), t_t(0x1.b546adb4fff9fp-36), t_t(0x1.fee514d605f47p-30), t_t(0x1.7d8aba38e94e5p-24), t_t(0x1.8dde8c8a5f14fp-19), t_t(0x1.333847cead132p-14), t_t(0x1.6e549758397a7p-10), t_t(0x1.5bf16c6d3c2e1p-6), t_t(0x1.0da0d5fc74909p-2), t_t(0x1.5b73be4c7df51p+1), t_t(0x1.79ff3dd485aafp+4), t_t(0x1.5f880142f6146p+7), t_t(0x1.1a5d623e94efbp+10), t_t(0x1.8b32414a15ad6p+12), t_t(0x1.e567c0b9ae06dp+14), t_t(0x1.073aa769c2856p+17), t_t(0x1.fad561fe4e26dp+18), t_t(0x1.b30d1de507946p+20), t_t(0x1.4e3715bb87823p+22), t_t(0x1.cd08828008dc2p+23), t_t(0x1.1e40be2e52dcbp+25), t_t(0x1.40b0751b1ac9ap+26), t_t(0x1.44af62e39e83bp+27), t_t(0x1.297beeddd88f6p+28), t_t(0x1.edc6fed97e94bp+28), t_t(0x1.73691a2219c14p+29), t_t(0x1.fa80b130a3bep+29), t_t(0x1.39067efdc7df4p+30), t_t(0x1.5e7b40d7d2084p+30), t_t(0x1.632168836b94bp+30), t_t(0x1.45306ea537b95p+30), t_t(0x1.0c9658d7625dap+30), t_t(0x1.8f363c6beaf6dp+29), t_t(0x1.0a211233648f8p+29), t_t(0x1.3d1140ddfc441p+28), t_t(0x1.4ff84500dc349p+27), t_t(0x1.3ac7eb2566ea2p+26), t_t(0x1.02e7a8f6f2da6p+25), t_t(0x1.72835dbeba284p+23), t_t(0x1.c807524049b3fp+21), t_t(0x1.db9f88c9c8139p+19), t_t(0x1.9c223405c5ca7p+17), t_t(0x1.20b217bb154afp+15), t_t(0x1.3a21976ecc423p+12), t_t(0x1.f222222222222p+8), t_t(0x1.fffffffffffffp+4), t_t(0x1p+0),
         t_t(0x1.af85c612db827p-57), t_t(0x1.14071f22622c9p-47), t_t(0x1.b76de43061b4dp-40), t_t(0x1.580345ea18a2fp-33), t_t(0x1.4345fe11a7d12p-27), t_t(0x1.97ce2963da7d9p-22), t_t(0x1.72db482ac78fp-17), t_t(0x1.fed78f075e975p-13), t_t(0x1.143897a044dcfp-8), t_t(0x1.e20006ceffea9p-5), t_t(0x1.5a9584bb14efbp-1), t_t(0x1.a1cf2ba29ec45p+2), t_t(0x1.ac133e6dc5f61p+5), t_t(0x1.79075d74bf151p+8), t_t(0x1.202d0722189ap+11), t_t(0x1.815a5aaf5cbdp+13), t_t(0x1.c5cf0d4a1b3e1p+15), t_t(0x1.d95a24c984d7p+17), t_t(0x1.b77264b4ff521p+19), t_t(0x1.6c9dd55c45173p+21), t_t(0x1.0f548849c01b7p+23), t_t(0x1.6b3fe036788fcp+24), t_t(0x1.b686c7648a51p+25), t_t(0x1.de5945ab46044p+26), t_t(0x1.d838ca4390d73p+27), t_t(0x1.a667011eb005dp+28), t_t(0x1.56a6590e161a5p+29), t_t(0x1.f860db64e3563p+29), t_t(0x1.50d975bcd883ap+30), t_t(0x1.982b9feae15cp+30), t_t(0x1.c067f15bc36a8p+30), t_t(0x1.be25dd43054b9p+30), t_t(0x1.9172518238c69p+30), t_t(0x1.460a927a4c2d2p+30), t_t(0x1.dcd3dd673fc23p+29), t_t(0x1.38f3eaee0d423p+29), t_t(0x1.6f49f5e97cbcbp+28), t_t(0x1.7f93d79a227a8p+27), t_t(0x1.6261b6cc99775p+26), t_t(0x1.1f8cecd0616f2p+25), t_t(0x1.96245e5a0034p+23), t_t(0x1.ed9126d862b89p+21), t_t(0x1.fc789601da07cp+19), t_t(0x1.b35cc9600b668p+17), t_t(0x1.2d73302a7a1f2p+15), t_t(0x1.4455895895897p+12), t_t(0x1.fcb60b60b60b5p+8), t_t(0x1.02aaaaaaaaaaap+5), t_t(0x1p+0),
         t_t(0x1.1cb745fc9b634p-63), t_t(0x1.115239ae67feep-52), t_t(0x1.6b7ba636e174fp-44), t_t(0x1.900ac15f1a6fcp-37), t_t(0x1.e61e11875743fp-31), t_t(0x1.797760e2ea70dp-25), t_t(0x1.992e4fde392f5p-20), t_t(0x1.48690db189c6p-15), t_t(0x1.97072f3b623a5p-11), t_t(0x1.91dac0ce947bfp-7), t_t(0x1.43bcdcd7b123p-3), t_t(0x1.b1ca1eae7d236p+0), t_t(0x1.ead9443697a62p+3), t_t(0x1.daf2ed75bd6fdp+6), t_t(0x1.8d1885b3be9cbp+9), t_t(0x1.2162cadd6e757p+12), t_t(0x1.725aa8b0181e3p+14), t_t(0x1.a2c9e16d34fe2p+16), t_t(0x1.a4a58d86d96d4p+18), t_t(0x1.790242bbe44b3p+20), t_t(0x1.2eabceedecfd8p+22), t_t(0x1.b4be0308357dep+23), t_t(0x1.1bf4459f01287p+25), t_t(0x1.4d7f4d5e6ee01p+26), t_t(0x1.626f043fbfbcep+27), t_t(0x1.555c40df388dep+28), t_t(0x1.2a43d3433a59dp+29), t_t(0x1.d935705b4afffp+29), t_t(0x1.54efb2e3fc61dp+30), t_t(0x1.be36081ade13dp+30), t_t(0x1.0920c124f225bp+31), t_t(0x1.1de0716f51a39p+31), t_t(0x1.17636c66580a9p+31), t_t(0x1.ee36e069c1971p+30), t_t(0x1.8acc0ef93238cp+30), t_t(0x1.1c20aa011f7cfp+30), t_t(0x1.6f467361277cbp+29), t_t(0x1.a8b41d92e49b1p+28), t_t(0x1.b53cb92764c43p+27), t_t(0x1.8e689a734faf9p+26), t_t(0x1.3efa5dcda7cefp+25), t_t(0x1.bcbb70d47edadp+23), t_t(0x1.0adc76d2e21e6p+22), t_t(0x1.0f97fa9c6767ap+20), t_t(0x1.cba07277a3994p+17), t_t(0x1.3a9f43e32d21dp+15), t_t(0x1.4ec16c16c16c3p+12), t_t(0x1.03b3333333333p+9), t_t(0x1.0555555555554p+5), t_t(0x1p+0),
         t_t(0x1.64fdae9aa72d9p-58), t_t(0x1.db7357eff669bp-49), t_t(0x1.89bb7e45eea39p-41), t_t(0x1.4086a85b12a95p-34), t_t(0x1.39189da31bb3p-28), t_t(0x1.9a75a2fed2e58p-23), t_t(0x1.83da21c4a46e3p-18), t_t(0x1.158b0e5ad60c8p-13), t_t(0x1.37da7aaf18488p-9), t_t(0x1.1aba6c274670ap-5), t_t(0x1.a68f4c354e33p-2), t_t(0x1.08c2d22724e46p+2), t_t(0x1.1a10d08af5a39p+5), t_t(0x1.0269e16027334p+8), t_t(0x1.9b13fbd539ffp+10), t_t(0x1.1e285e31a0a85p+13), t_t(0x1.5f0dd214ef3d1p+15), t_t(0x1.7db18c78a1191p+17), t_t(0x1.71a2c5e92c925p+19), t_t(0x1.402c0973b6dd8p+21), t_t(0x1.f1e3c4c6308a5p+22), t_t(0x1.5c911fd1d841fp+24), t_t(0x1.b89196e92f233p+25), t_t(0x1.f7bd672c14708p+26), t_t(0x1.04f6a9f9a9d9ep+28), t_t(0x1.eab083e840a7bp+28), t_t(0x1.a302bdaecae67p+29), t_t(0x1.4531d0ef0588dp+30), t_t(0x1.cae7a47d47e59p+30), t_t(0x1.265d831958952p+31), t_t(0x1.572f1c0eadc7bp+31), t_t(0x1.6b51895504a7dp+31), t_t(0x1.5ce0783ca717dp+31), t_t(0x1.2f63b9409924ap+31), t_t(0x1.dce39df1a1d44p+30), t_t(0x1.51dd9e976e23cp+30), t_t(0x1.ae2e9ea0cc233p+29), t_t(0x1.ea3c493a46a1dp+28), t_t(0x1.f1a1dcba27539p+27), t_t(0x1.bf4bbb56e72e8p+26), t_t(0x1.616bc0e93b085p+25), t_t(0x1.e67def5fc5d3cp+23), t_t(0x1.20528e766ac99p+22), t_t(0x1.21ed952e31065p+20), t_t(0x1.e4f5f2747c9d2p+17), t_t(0x1.4838a28a28a2ap+15), t_t(0x1.5965d75d75d77p+12), t_t(0x1.0919999999999p+9), t_t(0x1.07fffffffffffp+5), t_t(0x1p+0),
         t_t(0x1.cd9065a12ae59p-65), t_t(0x1.cd44b1fd26f86p-54), t_t(0x1.3f116cd4ecc7fp-45), t_t(0x1.6d19d8cbacebep-38), t_t(0x1.cd141584d684cp-32), t_t(0x1.73f7fe7cd7e41p-26), t_t(0x1.a2d79bb7ea306p-21), t_t(0x1.5d2604493f278p-16), t_t(0x1.c16e437b4b6eep-12), t_t(0x1.ccdcf13bd1ca7p-8), t_t(0x1.81aa8157c91c5p-4), t_t(0x1.0c72f2a5e9043p+0), t_t(0x1.3bab7e74773bep+3), t_t(0x1.3d8524e1dfc24p+6), t_t(0x1.141263671f90cp+9), t_t(0x1.a2a0cd49fef97p+11), t_t(0x1.16d6a5253dd15p+14), t_t(0x1.4864b100dd036p+16), t_t(0x1.57c3b86686327p+18), t_t(0x1.415481c3826ccp+20), t_t(0x1.0d42cce3b2f32p+22), t_t(0x1.95e4d3e71612cp+23), t_t(0x1.13f5167effddbp+25), t_t(0x1.53474a3a6875p+26), t_t(0x1.79e6f32c458d9p+27), t_t(0x1.7df176771848cp+28), t_t(0x1.5eb565e844635p+29), t_t(0x1.24d2b65cf3917p+30), t_t(0x1.bce1b12a03189p+30), t_t(0x1.3389edf7b2377p+31), t_t(0x1.82e2fa91af622p+31), t_t(0x1.baa8d5497d8dep+31), t_t(0x1.cc43631ab1511p+31), t_t(0x1.b26281538a5c2p+31), t_t(0x1.73834bafc2f24p+31), t_t(0x1.1f57adf867a5bp+31), t_t(0x1.90e9042401afbp+30), t_t(0x1.f6e676d8ecc15p+29), t_t(0x1.1a750412309fap+29), t_t(0x1.1ac1c58e17eb1p+28), t_t(0x1.f583aa3abd3bep+26), t_t(0x1.87211a5a027d8p+25), t_t(0x1.09d225d4f57c5p+24), t_t(0x1.373ef4f30d3e9p+22), t_t(0x1.354838867d9c6p+20), t_t(0x1.ff663d6063627p+17), t_t(0x1.5641a21693306p+15), t_t(0x1.644362e0d8b84p+12), t_t(0x1.0e8e38e38e38ep+9), t_t(0x1.0aaaaaaaaaaa9p+5), t_t(0x1p+0),
         t_t(0x1.273634790eb37p-59), t_t(0x1.98fbd47cb349dp-50), t_t(0x1.60179ebf791bbp-42), t_t(0x1.29d4167024d43p-35), t_t(0x1.2e2f8e5d85e5dp-29), t_t(0x1.9b63438828c74p-24), t_t(0x1.939c9126e9eb2p-19), t_t(0x1.2bdb196b127cdp-14), t_t(0x1.5dcc72f28fd3ap-10), t_t(0x1.49458bee0a544p-6), t_t(0x1.ff0970509edc1p-3), t_t(0x1.4c928ef7d1376p+1), t_t(0x1.7019edda2a37ap+4), t_t(0x1.5e7a6234b3a45p+7), t_t(0x1.21d3b20cc13e7p+10), t_t(0x1.a3b51b23ba7e3p+12), t_t(0x1.0bebeef210d1dp+15), t_t(0x1.2f565b2dcddfcp+17), t_t(0x1.3216a7a70aa36p+19), t_t(0x1.147622f7774cep+21), t_t(0x1.c0a7bc4626ebdp+22), t_t(0x1.481487f9e33cdp+24), t_t(0x1.b18eb496933ddp+25), t_t(0x1.036d5527cde4p+27), t_t(0x1.19ab0a12f8feap+28), t_t(0x1.15dab153fe1cep+29), t_t(0x1.f29e0e6c86c68p+29), t_t(0x1.9742179ba3fb1p+30), t_t(0x1.2ef079112fe9ap+31), t_t(0x1.9a81257b8e092p+31), t_t(0x1.fa94b42def64dp+31), t_t(0x1.1c8238df52dcfp+32), t_t(0x1.22a19730e8f5cp+32), t_t(0x1.0da94ea6aff0cp+32), t_t(0x1.c5c4bc9b2946dp+31), t_t(0x1.5977a95990d71p+31), t_t(0x1.daba7bc13a17fp+30), t_t(0x1.25682347d079fp+30), t_t(0x1.44f14d9499cd5p+29), t_t(0x1.40da6f56bd9f8p+28), t_t(0x1.18c98836bf677p+27), t_t(0x1.b05eed2d7da04p+25), t_t(0x1.223519901a03fp+24), t_t(0x1.4fb6abe45f583p+22), t_t(0x1.49b36053c165fp+20), t_t(0x1.0d7d3c56ee4a3p+18), t_t(0x1.64bc9ed1bf993p+15), t_t(0x1.6f5aa6550ffbbp+12), t_t(0x1.1411111111111p+9), t_t(0x1.0d55555555554p+5), t_t(0x1p+0),
        };

      auto cotderiv = [&coefs](int n, T x){
        using elt_t = eve::underlying_type_t<T>;
        auto pi = eve::pi(eve::as<elt_t>());
        if (n == 0)
        {
          return pi*eve::cotpi(x);
        }
        else if (n < 100)
        {
          auto siz = [](auto i){
            //size of the nth polynomial coefficients
            return std::size_t(eve::floor(0.5*(i+1))+1);
          };
          auto beg = [](int i){
            // \f$\sum_(i=0)^(n - 1)(floor((i + 1)/2) + 1)\f$
            // index of the first element of the ith polynomial coefficients
            auto h = 0.5;
            auto c =  eve::ceil(h*(3-i));
            auto d =  eve::floor(h*i);
            return int(h*(c*c -3*c +d*d + d + 2*i))+1;
          };
          auto ps =  std::span(&coefs[beg(n)], siz(n));
          auto cx = eve::cotpi(x);
          auto val = eve::reverse_horner(cx*cx, ps);
          return (eve::is_odd(n) ? val : val*cx)*eve::pow(pi, n+1);
        }
        else
        {
          auto p = n+1;
          x -= eve::nearest(x);
          auto s = eve::rec(eve::pow(x, p));
          n = 1;
          auto s0 = eve::zero(eve::as(s));
          while (1)
          {
            s0 = s;
            auto a = eve::pow(x+n, p);
            auto b = eve::pow(x-n, p);
            auto t = (a + b) / (a * b);
            s += t;
            if (eve::all(s == s0)) break;
            n += 1;
          }
          return s;
        }
      };

      using u_t =  eve::underlying_type_t<T>;
      int s = m+1;
      auto mg = -eve::tgamma(u_t(s));
      auto sa =  eve::sign_alternate(m);
      auto gez = eve::is_positive(z);

      auto br_pos =  [gez, s, mg, sa](auto zz){
        zz = if_else(gez, zz, eve::zero);
        return sa*eve::hurwitz(s, zz)*mg;
      };

      auto br_neg =  [gez, s, mg, sa, m, cotderiv](auto zz){
        zz = if_else(gez, eve::mzero(eve::as(zz)), zz);
        return (eve::hurwitz(s,eve::oneminus(zz))+sa*cotderiv(m, zz))*mg;
      };


      auto notdone = is_not_flint(z) || gez;
      auto r = inf(as<T>()); // negative flint z elements

      if( eve::any(notdone) )
      {
        notdone = next_interval(br_pos, notdone, gez, r, z);
        if( eve::any(notdone) )
        {
          last_interval(br_neg, notdone, r, z);
        }
      }
      return r;
    }
  }
}
